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Let \( G \) be a connected graph and let \( U \) be a set of vertices in \( G \). A \emph{minimal \( U \)-tree} is a subtree \( T \) of \( G \) that contains \( U \) and has the property that every vertex of \( V(T) – U \) is a cut-vertex of \( \langle V(T) \rangle \). The \emph{monophonic interval} of \( U \) is the collection of all vertices of \( G \) that lie on some minimal \( U \)-tree. A set \( S \) of vertices of \( G \) is \( m_k \)-\emph{convex} if it contains the monophonic interval of every \( k \)-subset \( U \) of vertices of \( S \). Thus \( S \) is \( m_2 \)-convex if and only if it is \( m \)-convex.
In this paper, we consider three local convexity properties with respect to \( m_3 \)-convexity and characterize the graphs having either property.